Solve 4/11sqrt{77/8}

$$\frac { 4 } { 11 } \sqrt { \frac { 77 } { 8 } }$$

Evaluate

$$\frac{\sqrt{154}}{11}\approx 1.12815215$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{77}{8}} as the division of square roots \frac{\sqrt{77}}{\sqrt{8}}.

$$\frac{4}{11}\times \frac{\sqrt{77}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{4}{11}\times \frac{\sqrt{77}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{77}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{4}{11}\times \frac{\sqrt{77}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4}{11}\times \frac{\sqrt{77}\sqrt{2}}{2\times 2}$$

To multiply \sqrt{77} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{4}{11}\times \frac{\sqrt{154}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{4}{11}\times \frac{\sqrt{154}}{4}$$

Multiply \frac{4}{11} times \frac{\sqrt{154}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{154}}{11\times 4}$$

Cancel out 4 in both numerator and denominator.

$$\frac{\sqrt{154}}{11}$$